Maps preserving the inner local spectral radius zero of generalized product of operators

Abstract

Let ${\mathscr {B}}(X)$ be the algebra of all bounded linear operators on a complex Banach space X. For an operator $T\in {\mathscr {B}}(X)$, let $\iota _{T}(x)$ denote the inner local spectral radius of T at any vector $x\in X$. For an integer $k\ge 2$, let $(i_1,\dots ,i_m)$ be a finite sequence such that $\{i_1,\dots ,i_m\}=\{1,\dots ,k\}$ and at least one of the terms in $(i_1,\dots ,i_m)$ appears exactly once. The generalized product of k operators $T_1,\dots ,T_k\in {\mathscr {B}}(X)$ is defined by

$$\begin{aligned} T_1*\cdots *T_k:=T_{i_1}T_{i_2}\dots T_{i_m}, \end{aligned}$$

and includes the usual product TS and the triple product TST. We show that a surjective map $\varphi$ on ${\mathscr {B}}(X)$ satisfies

$$\begin{aligned} \iota _{_{\varphi (T_1)*\cdots *\varphi (T_k)}}(x)=0 \Longleftrightarrow \iota _{ _{T_{1}*\cdots *T_{k}}}(x) = 0 \end{aligned}$$

for all $x\in X$ and all $T_{1},\ldots ,T_{k} \in {\mathscr {B}}(X)$ if and only if there exists a map $\gamma : {\mathscr {B}}(X)\rightarrow {\mathbb {C}}\setminus \{0\}$ such that $\varphi (T)=\gamma (T) T$ for all $T\in {\mathscr {B}}(X)$.